Artin formalism for Selberg zeta functions of co-finite Kleinian groups

Eliot Brenner[1]; Florin Spinu[2]

  • [1] University of Minnesota 206 Church Street SE Minneapolis USA, 55455
  • [2] Johns Hopkins University Baltimore USA, 21218

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 1, page 59-75
  • ISSN: 1246-7405

Abstract

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Let Γ ∖ ℍ 3 be a finite-volume quotient of the upper-half space, where Γ ⊂ SL ( 2 , ℂ ) is a discrete subgroup. To a finite dimensional unitary representation χ of Γ one associates the Selberg zeta function Z ( s ; Γ ; χ ) . In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if Γ ˜ is a finite index group extension of Γ in SL ( 2 , ℂ ) , and π = Ind Γ Γ ˜ χ is the induced representation, then Z ( s ; Γ ; χ ) = Z ( s ; Γ ˜ ; π ) . In the second part of the paper we prove by a direct method the analogous identity for the scattering function, namely φ ( s ; Γ ; χ ) = φ ( s ; Γ ˜ ; π ) , for an appropriate normalization of the Eisenstein series.

How to cite

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Brenner, Eliot, and Spinu, Florin. "Artin formalism for Selberg zeta functions of co-finite Kleinian groups." Journal de Théorie des Nombres de Bordeaux 21.1 (2009): 59-75. <http://eudml.org/doc/10876>.

@article{Brenner2009,
abstract = {Let $\Gamma \backslash \mathbb\{H\}^3$ be a finite-volume quotient of the upper-half space, where $\Gamma \subset \{\rm SL\}(2,\mathbb\{C\})$ is a discrete subgroup. To a finite dimensional unitary representation $\chi $ of $\Gamma $ one associates the Selberg zeta function $Z(s;\Gamma ;\chi )$. In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if $\tilde\{\Gamma \}$ is a finite index group extension of $\Gamma $ in $\{\rm SL\}(2,\mathbb\{C\})$, and $\pi =\{\rm Ind\}_\{\Gamma \}^\{\tilde\{\Gamma \}\}\chi $ is the induced representation, then $Z(s;\Gamma ;\chi )=Z(s;\tilde\{\Gamma \};\pi )$. In the second part of the paper we prove by a direct method the analogous identity for the scattering function, namely $\phi (s;\Gamma ;\chi )=\phi (s;\tilde\{\Gamma \};\pi )$, for an appropriate normalization of the Eisenstein series.},
affiliation = {University of Minnesota 206 Church Street SE Minneapolis USA, 55455; Johns Hopkins University Baltimore USA, 21218},
author = {Brenner, Eliot, Spinu, Florin},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Artin Formalism; Selberg Zeta function; Kleinian groups; Fuchsian groups hyperbolic 3-manifolds; scattering matrix; Eisenstein series; Artin formalism; Fuchsian groups; hyperbolic 3-manifolds},
language = {eng},
number = {1},
pages = {59-75},
publisher = {Université Bordeaux 1},
title = {Artin formalism for Selberg zeta functions of co-finite Kleinian groups},
url = {http://eudml.org/doc/10876},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Brenner, Eliot
AU - Spinu, Florin
TI - Artin formalism for Selberg zeta functions of co-finite Kleinian groups
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 1
SP - 59
EP - 75
AB - Let $\Gamma \backslash \mathbb{H}^3$ be a finite-volume quotient of the upper-half space, where $\Gamma \subset {\rm SL}(2,\mathbb{C})$ is a discrete subgroup. To a finite dimensional unitary representation $\chi $ of $\Gamma $ one associates the Selberg zeta function $Z(s;\Gamma ;\chi )$. In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if $\tilde{\Gamma }$ is a finite index group extension of $\Gamma $ in ${\rm SL}(2,\mathbb{C})$, and $\pi ={\rm Ind}_{\Gamma }^{\tilde{\Gamma }}\chi $ is the induced representation, then $Z(s;\Gamma ;\chi )=Z(s;\tilde{\Gamma };\pi )$. In the second part of the paper we prove by a direct method the analogous identity for the scattering function, namely $\phi (s;\Gamma ;\chi )=\phi (s;\tilde{\Gamma };\pi )$, for an appropriate normalization of the Eisenstein series.
LA - eng
KW - Artin Formalism; Selberg Zeta function; Kleinian groups; Fuchsian groups hyperbolic 3-manifolds; scattering matrix; Eisenstein series; Artin formalism; Fuchsian groups; hyperbolic 3-manifolds
UR - http://eudml.org/doc/10876
ER -

References

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  1. E. Brenner, F. Spinu, Artin Formalism, for Kleinian Groups, via Heat Kernel Methods. Submitted to Serge Lang Memorial Volume. Zbl1233.11057
  2. P. Cohen, P. Sarnak, Lecture notes on Selberg trace formula (unpublished). 
  3. J. Elstrodt, F. Grunewald, J. Mennicke, Groups acting on hyperbolic space. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. Zbl0888.11001MR1483315
  4. J. Friedman, The Selberg trace formula and Selberg-zeta function for cofinite Kleinian groups with finite-dimensional unitary representations. Math. Zeit. 50 (2005), No.4. Zbl1135.11026MR2180383
  5. J. Friedman, Analogues of the Artin factorization formula for the automorphic scattering matrix and Selberg zeta-function associated to a Kleinian group. Arxiv:math/0702030. MR2415021
  6. R. Gangolli, G. Warner, Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one. Nagoya Math. J. 78 (1980), 1–44. MR571435
  7. J. Jorgenson, S. Lang, Artin formalism and heat kernels. Jour. Reine. Angew. Math. 447 (1994), 165–280. Zbl0789.11055MR1263173
  8. A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric spaces with applications to Dirichlet series. J. Indian Math. Soc. 20 (1956), 47–87. Zbl0072.08201MR88511
  9. A.B. Venkov, The Artin Takagi formula for Selberg’s zeta-function and the Roelcke conjecture. Soviet Math. Dokl. 20 (1979), No.4, 745–748. Zbl0432.30036
  10. A. B. Venkov, Spectral Theory of Automorphic Functions. Proceedings of the Steklov Institute of Mathematics 4, 1982. Zbl0501.10029MR692019
  11. A. B. Venkov, P. Zograf, Analogues of Artin’s factorization in the spectral theory of automorphic functions. Math. USSR Izvestiya 2 (1983), No. 3, 435–443. Zbl0527.10020

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